I'm very confused about this problem. I understand that to show {A_1, ..., A_4} is a basis, I must show 1.) the set is linearly independent, and 2.) it is a spanning set; however, I know there is a less complicated way instead of going through these 2 steps. I'm really not sure what the "easy" way is for doing this problem... it hints that there is a Thm that will help solve the problem, but I have found none that fit the bill. Can someone please help me with all this?

A basis, for a vector space of dimension n, satisfies 3 properties:
1) it spans the space
2) it is independent
3) it contains n vectors.

If any two of those are true the third must be. If you know, or are given, that M_2 R is a 4 dimensional vector space, that might be the theorem you are referring to. Since this set contains 4 matrices, (3) above is clearly satisfied and you only need to prove either (1) or (2), not both.

If you do not know that the M_2 R is 4 dimensional, then you will need to prove both (1) and (2), and, honestly, they are not that difficult! Both reduce to 4 equations in 4 unknowns and you really only need to show that those equations have a unique solution- which is true if the 4 by 4 coefficient matrix does not have 0 determinant. It would probably take you less time than waiting for a response on the internet.